Note that \(\Phi ({\mathbf {p}}, t_1, t_2, v_\theta )\)  can represent both forward (\(t_2> t_1\) ) or backward motion (\(t_2< t_1\) ) given the initial location and velocity-field. To solve the flow-field for discrete video time-steps, we approximate the above continuous integral equation using a neural-ODE solver [1]}.
